The connection between factors and zeros is pretty simple, the zeros that you find from an equation is the factored function of that same equation. For example in the equation above, the zeros that were found are the numbers used in the factored form of the polynomial. Division helps us factor polynomials because it is a quicker way to find the solutions. The degree of the polynomial helps to predict the number of zeros by telling us in the highest degree, in the polynomial above the highest degree is 4, therefore that polynomial should have 4 zeros. Even though it does not look like it this polynomial has 4 zeros. There is one repeating zero in this equation which you would find out when using division (another reason division helps us factor polynomials). The highest power of the degree may not always tell us the number of factors because, like the example, there could be a repeated zero. Also it may not give the number of factors because it could be something that gives you an "ugly" square root or an imaginary solution.
The connection between factors and zeros is pretty simple, the zeros that you find from an equation is the factored function of that same equation. For example in the equation above, the zeros that were found are the numbers used in the factored form of the polynomial. Division helps us factor polynomials because it is a quicker way to find the solutions. The degree of the polynomial helps to predict the number of zeros by telling us in the highest degree, in the polynomial above the highest degree is 4, therefore that polynomial should have 4 zeros. Even though it does not look like it this polynomial has 4 zeros. There is one repeating zero in this equation which you would find out when using division (another reason division helps us factor polynomials). The highest power of the degree may not always tell us the number of factors because, like the example, there could be a repeated zero. Also it may not give the number of factors because it could be something that gives you an "ugly" square root or an imaginary solution.
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From the Even and Odd Functions activity I learned that not all even degree functions are necessarily even functions and not all odd degree functions are necessarily odd function. The rule to determine if a function is even is if and only if it follows the pattern f(-x) = f(x). The rule to determine if it is odd is that it has to follow f(-x) = -f(x). They are similar because they both have some symmetry to their graphs and show a pattern. They are different because odd functions y values change sign while even functions do not. You check if a function is odd by putting a negative x into the equation and seeing the outcome if it is the same as f(x) it is even if it is the same as –f(x) it is odd. If a function is quadratic it will always be even. But there is no always odd function group. I would like a little more clarification on what functions belong to which groups; it's still a little fuzzy.
To get this function I looked at all the past graphs that I had already found the equations for and put them all in one equation. But the fourth graph only showed pieces of each graph so I knew I had to limit the domain for each piece. So I used the format of a piecewise function and I limited it when it looked like the graph changed. I figured that since the graph was y=-3x+2 until x was 0 then I would put it as "if x < 0". I used this reasoning to figure out each piece of the function. The domain is the first coordinate from the first piece of the function and the last coordinate is the same as the last coordinate of the domain of the third piece of the function. I found the range by simply looking at the graph. It ended up being the same as graph 3 and the first coordinate was the same as graph 2, while the last coordinate was the same as graph 1.
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September 2015
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